#8417077

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3643 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 64 points:

(1,2,5,8)(3,6,10,13)(4,7,11,14)(9,12,15,16)(17,18)(19,27)(20,31)(21,32)(22,26)(23,24)(25,28)(29,30)(33,34,39,43)(35,40,44,37)(36,47,48,41)(38,42,54,58)(45,51,56,49)(46,55,53,52)(50,60,61,57)(59,64,63,62);(1,3,5,10)(2,6,8,13)(4,9,11,15)(7,12,14,16)(18,24)(19,28)(20,32)(26,30)(35,44)(36,48)(37,40)(41,47)(46,53)(50,61)(52,55)(57,60);(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,19,23,28)(18,25,24,27)(20,29,32,22)(21,30,31,26)(33,35)(34,40)(36,45)(37,43)(38,53)(39,44)(41,49)(42,52)(46,54)(47,51)(48,56)(50,63)(55,58)(57,64)(59,61)(60,62);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,15)(14,16)(17,20)(18,21)(19,29)(22,28)(23,32)(24,31)(25,30)(26,27)(33,36,39,48)(34,41,43,47)(35,45,44,56)(37,51,40,49)(38,50,54,61)(42,57,58,60)(46,59,53,63)(52,64,55,62);(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,21)(18,20)(19,30)(22,27)(23,31)(24,32)(25,29)(26,28)(33,37,39,40)(34,35,43,44)(36,49,48,51)(38,52,54,55)(41,56,47,45)(42,46,58,53)(50,62,61,64)(57,63,60,59);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,12,13,16)(17,22,23,29)(18,26,24,30)(19,20,28,32)(21,27,31,25)(33,38)(34,42)(35,46)(36,50)(37,52)(39,54)(40,55)(41,57)(43,58)(44,53)(45,59)(47,60)(48,61)(49,62)(51,64)(56,63)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324273);